A General History Matching Algorithm for Three-Phase, Three-Dimensional Petroleum Reservoirs

Abstract

An algorithm is developed to estimate absolute permeability in multi-phase, multilayered petroleum reservoirs. Summary A numerical algorithm is developed to estimate absolute permeability in multi-phase, multilayered petroleum reservoirs permeability in multi-phase, multilayered petroleum reservoirs based upon noisy observation data, such as pressure, water cut, gas-oil ratio and rates of liquid and gas production from individual layers. A Chevron Black-Oil code is used as the basic reservoir simulator in conjunction with this history matching algorithm. Since the history matching inverse problem is ill-posed due to its large dimensionality and the insensitivity of the permeability to measured well data, regularization and spline approximation of the spatially varying absolute permeability are employed to render the problem computationally well behaved. A stabilizing functional problem computationally well behaved. A stabilizing functional with a gradient operator is used to measure the non-smoothness of the parameter estimates in the regularization approach, and the regularization parameter is determined automatically during the computation. The numerical minimization algorithm is based on the partial conjugate gradient method of Nazareth. Numerical examples partial conjugate gradient method of Nazareth. Numerical examples are considered in two- and three-phase reservoirs with three layers. The effects of the degree of regularization, spline approximation versus zonation, and differing true areal permeability distributions on the performance of the method are considered. Introduction Estimation of the properties of multilayered, multiphase reservoirs remains an important problem in reservoir analysis. The knowledge of these properties, such as absolute permeability, forms the basis for determining an optimal strategy of oil recovery. The process of estimating unknown properties in a mathematical process of estimating unknown properties in a mathematical reservoir model to give the best fit to measure well data is called history matching. The history matching problem is a notoriously difficult one for several reasons:(1)In a reservoir. properties vary with location; thus, conceptually an infinite number of parameters are required for a full description of the reservoir. Computationally, a reservoir simulator contains a finite number of parameters corresponding to the number of grid blocks, and in field-scale simulations, a simulator may contain on the order of 10,000 grid blocks.(2)The history matching problem is theoretically imposed, which means that small instabilities in the data can lead to large perturbations in the estimated parameters.(3)Many actual perturbations in the estimated parameters.(3)Many actual reservoirs involve significant vertical as well as horizontal property variations, requiring the estimation of property property variations, requiring the estimation of property distributions in both directions.(3)History matching situations may involve full threephase (oil, water and gas) behavior, wherein traditional observations at wells, such as pressures and water-oil ratio, may not be adequate to enable property estimation; the sorts of observation data necessary have not been addressed previously in the literature. There exist a number of prior papers on automatic history matching, which have been organized according to reservoir dimensionality and number of phases considered in Table 1. The goal of the present work is to develop a general history matching code applicable for three-dimensional, three-phase reservoirs. To alleviate the ill-conditioning in the history matching problem, several approaches have been tried, such as decreasing problem, several approaches have been tried, such as decreasing the number of parameters to be estimated and, in addition, utilizing any available information to constrain the choice of the unknown parameters. One way of reducing the number of parameters is to parameters. One way of reducing the number of parameters is to divide the reservoir into a relatively small number of zones and to assume that the properties are uniform within each zone. While this approach is effective in reducing the number of unknowns, sufficient a priori information is not usually available to enable specification of the zones on any physical basis. A modification to zonation is to use prior information expressed as an assumed probability distribution for the zonal reservoir properties. If probability distribution for the zonal reservoir properties. If certain a priori knowledge is assumed about the mean values and correlations of the parameters, the history matching performance index can be modified to include a term that penalizes the weighted deviations of the parameters from their assumed mean values. Although it has been shown that better-conditioned estimates may be obtained when a priori statistical information is used, sufficient knowledge of the nature of the unknown parameters is not generally available to specify the parameters needed to carry out such a Bayesian estimation.

Estimation of plume distribution for carbon sequestration using parameter estimation with limited monitoring data

“…We acknowledge the work of Oliver and Chen [2011] for part of this literature search. An overview of algorithms and applications can be found in Makhlouf et al [1993] and a more recent overview in the book from Oliver et al [2008].…”

Mapping of Reservoir Properties and Facies Through Integration of Static and Dynamic Data

“…where (Makhlouf et al, 1993) or variable metric method (Yang and Watson, 1988), we need only compute the gradient of the objective function and this can be done by setting β = O(m) in the adjoint procedure. In this case, one only needs to solve the adjoint system Eq.…”

“…Here, we use the Polak-Ribière form of the nonlinear conjugate gradient algorithm; see Fletcher (1987). Makhlouf et al (1993) applied the conjugate gradient method to estimate 450 gridblock permeabilities by history matching production data obtained under multiphase flow conditions. In one example, 110 iterations were required to obtain conver-gence.…”

Recent progress on reservoir history matching: a review

“…In the early 1990s, the state-of-the-art in automatic history matching was probably best typified by the work of Makhlouf et al [114] who demonstrated a methodology capable of history matching synthetic data from two-and three-phase reservoirs (simulation grid and well locations are shown in Fig. 2a).…”

“…2a). In Makhlouf et al [114], the permeability distribution is modeled by bicubic spline function, an adjoint system was used to compute the gradient, and a conjugate gradient method was used to adjust the coefficient of the spline function of each simulation gridblock (450 parameters in total) for minimization of the data mismatch. In the key example, the squared data mismatch function for 10 years of simulated history, including 1,413 pressure data, 785 water cut and rate data, was reduced by a factor of 200 in approximately 200 iterations, requiring 11 cpu-hours on a Cray XMP-48 with four CPU.…”